Modeling the Lips and Mouthpiece

Figure 8.1 depicts a schematic of a brass-player's lips positioned within a mouthpiece. The air pressure^9.2 is denoted inside the mouth, between the lips, and within the mouthpiece cup.

A simple first approximation is to assume that the air flow through the lips is inviscid (``frictionless'') and incompressible. In this case, the pressure between the lips is governed by Bernoulli's equation (see §F.5.6):

where is the air density (mass per unit volume) and denotes the particle velocity (see §F.5.1) (distance per unit time) between the lips. The particle velocity in the mouth is so small that it is usually neglected (approximated by zero).

Following Cullen et al. [98] and previous researchers [162,164], one may assume that the flow between the lips creates a jet in the mouthpiece having velocity . The pressure within the jet is therefore , the same as between the lips. The jet is assumed to mix turbulently with the air in the mouthpiece so that it dissipates its energy without raising pressure in the mouthpiece.^9.3

One- and two-mass models were also developed by Rodet and Vergez [394,545]. An early one-mass, ``swinging door'' model that works surprisingly well is in Cook's HosePlayer [89] and and TBone [88]. It is also used in the Brass.cpp patch in the Synthesis Tool Kit [91]; in this simplified model, the lip-valve is modeled as a second-order resonator whose output is squared and hard-clipped to a maximum magnitude.

The acoustic tube in a horn can be modeled as a simple